Theorem isstruct2 17066

Description: The property of being a structure with components in (1^st ‘𝑋)...(2^nd ‘𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.)

Assertion

Ref	Expression
isstruct2	⊢ (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Proof of Theorem isstruct2

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brstruct 17065	. . 3 ⊢ Rel Struct
2	1	brrelex12i 5674	. 2 ⊢ (𝐹 Struct 𝑋 → (𝐹 ∈ V ∧ 𝑋 ∈ V))
3		ssun1 4127	. . . . 5 ⊢ 𝐹 ⊆ (𝐹 ∪ {∅})
4		undif1 4425	. . . . 5 ⊢ ((𝐹 ∖ {∅}) ∪ {∅}) = (𝐹 ∪ {∅})
5	3, 4	sseqtrri 3979	. . . 4 ⊢ 𝐹 ⊆ ((𝐹 ∖ {∅}) ∪ {∅})
6		simp2 1137	. . . . . . 7 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → Fun (𝐹 ∖ {∅}))
7	6	funfnd 6518	. . . . . 6 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∖ {∅}) Fn dom (𝐹 ∖ {∅}))
8		elinel2 4151	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 ∈ (ℕ × ℕ))
9		1st2nd2 7966	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (ℕ × ℕ) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
10	8, 9	syl 17	. . . . . . . . . 10 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
11	10	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
12	11	fveq2d 6832	. . . . . . . 8 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (...‘𝑋) = (...‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
13		df-ov 7355	. . . . . . . . 9 ⊢ ((1st ‘𝑋)...(2nd ‘𝑋)) = (...‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
14		fzfi 13885	. . . . . . . . 9 ⊢ ((1st ‘𝑋)...(2nd ‘𝑋)) ∈ Fin
15	13, 14	eqeltrri 2828	. . . . . . . 8 ⊢ (...‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩) ∈ Fin
16	12, 15	eqeltrdi 2839	. . . . . . 7 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (...‘𝑋) ∈ Fin)
17		difss 4085	. . . . . . . . 9 ⊢ (𝐹 ∖ {∅}) ⊆ 𝐹
18		dmss 5847	. . . . . . . . 9 ⊢ ((𝐹 ∖ {∅}) ⊆ 𝐹 → dom (𝐹 ∖ {∅}) ⊆ dom 𝐹)
19	17, 18	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐹 ∖ {∅}) ⊆ dom 𝐹
20		simp3 1138	. . . . . . . 8 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom 𝐹 ⊆ (...‘𝑋))
21	19, 20	sstrid 3941	. . . . . . 7 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom (𝐹 ∖ {∅}) ⊆ (...‘𝑋))
22	16, 21	ssfid 9159	. . . . . 6 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom (𝐹 ∖ {∅}) ∈ Fin)
23		fnfi 9093	. . . . . 6 ⊢ (((𝐹 ∖ {∅}) Fn dom (𝐹 ∖ {∅}) ∧ dom (𝐹 ∖ {∅}) ∈ Fin) → (𝐹 ∖ {∅}) ∈ Fin)
24	7, 22, 23	syl2anc 584	. . . . 5 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∖ {∅}) ∈ Fin)
25		p0ex 5324	. . . . 5 ⊢ {∅} ∈ V
26		unexg 7682	. . . . 5 ⊢ (((𝐹 ∖ {∅}) ∈ Fin ∧ {∅} ∈ V) → ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V)
27	24, 25, 26	sylancl 586	. . . 4 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V)
28		ssexg 5263	. . . 4 ⊢ ((𝐹 ⊆ ((𝐹 ∖ {∅}) ∪ {∅}) ∧ ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐹 ∈ V)
29	5, 27, 28	sylancr 587	. . 3 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝐹 ∈ V)
30		elex 3457	. . . 4 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 ∈ V)
31	30	3ad2ant1 1133	. . 3 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝑋 ∈ V)
32	29, 31	jca 511	. 2 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∈ V ∧ 𝑋 ∈ V))
33		simpr 484	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
34	33	eleq1d 2816	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ 𝑋 ∈ ( ≤ ∩ (ℕ × ℕ))))
35		simpl 482	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑓 = 𝐹)
36	35	difeq1d 4074	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑓 ∖ {∅}) = (𝐹 ∖ {∅}))
37	36	funeqd 6509	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (Fun (𝑓 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})))
38	35	dmeqd 5850	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → dom 𝑓 = dom 𝐹)
39	33	fveq2d 6832	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (...‘𝑥) = (...‘𝑋))
40	38, 39	sseq12d 3963	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (dom 𝑓 ⊆ (...‘𝑥) ↔ dom 𝐹 ⊆ (...‘𝑋)))
41	34, 37, 40	3anbi123d 1438	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥)) ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
42		df-struct 17064	. . 3 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
43	41, 42	brabga 5477	. 2 ⊢ ((𝐹 ∈ V ∧ 𝑋 ∈ V) → (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
44	2, 32, 43	pm5.21nii 378	1 ⊢ (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3894   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897  ∅c0 4282  {csn 4575  ⟨cop 4581   class class class wbr 5093   × cxp 5617  dom cdm 5619  Fun wfun 6481   Fn wfn 6482  ‘cfv 6487  (class class class)co 7352  1^st c1st 7925  2^nd c2nd 7926  Fincfn 8875   ≤ cle 11153  ℕcn 12131  ...cfz 13413   Struct cstr 17063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-pre-mulgt0 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-sub 11352  df-neg 11353  df-nn 12132  df-n0 12388  df-z 12475  df-uz 12739  df-fz 13414  df-struct 17064

This theorem is referenced by:  structn0fun  17068  isstruct  17069  setsstruct2  17091